Thank you for getting in touch with your question. When you disable the password page that is a feature of all Shopify-made themes, no notifications will be sent to your customers. Some merchants may disable and re-enable the password page prior to the store's actual launch, so an automated notification would not be appropriate in this situation.
However, if you have had customers register an interest in your business prior to launch, you may wish to send an email out to them notifying them that the site is live. This email could even be coupled with some sort of promotional discount to help drive traffic to your store and receive sales shortly after launch. There are a variety of apps that integrate with Shopify that you could use to send marketing emails to your customers, such as Omnisend and Privy. You can take a look at other options here as well.
Finally, I'd also recommend checking out our guide on how to run successful email marketing campaigns on the Shopify blog, as well as our comprehensive pre-launch checklist. Best of luck with the launch, and if you have any further questions, please let me know!
Victor | Shopify Social Care